The inverse scattering method is applied to integration of the gravitation equations when the metric tensor depends on two coordinates only. The simplest soliton solutions are calculated explicitly.
The authors describe a practical method (equivalent to the inverse scattering problem technique), allowing one to obtain explicitly large classes of new exact solutions of the vacuum Einstein equations for the case when the metric tensor depends only on two variables, if simple particular solutions of the equations are known. Moreover, if developed further, the method allows one in principle to approach the problem of finding, in a certain sense, `all the solutions' of the equations of gravity for the two-dimensional case under consideration, and may lead to a solution of the corresponding Cauchy problem.
The L-A pair corresponding to the Einstein-Maxwell equations has been determined and the N soliton solutions have been constructed for the case when the space-time metric and the 4-potential of the electromagnetic field depend only on two coordinates.
It will be described here an application of inverse scattering problem ideas for electrovacuum case. (From paragraph four.)
The determinantal method of writing soliton solutions proposed here apparently has a simpler structure of the matrices of the determinants (than that of Neugebauer and Kramer), and, moreover, makes it possible to compute directly all the components of the metric without expressing them in terms of the complex Ernst potentials. [From footnote #1.]
We shall review some tools that have proved useful in the construction of supergravity theories: duality transformations (discrete and continuous) and their importance for the enlargement of the symmetry groups, the structure of generalized s-models where the scalar fields take their values on coset spaces G/H and their possible Lagrangians. The second topic will be a presentation of the disintegration triangle and its mysterious rules: it brings together the E7 and SU(8) invariances of n = 8 supergravity and the Ehlers symmetry of stationary solutions of pure Einstein theory. Most striking is the analogy between internal and space-time dimensions. Finally, five generalizations of the infinite dimensional Lie groups of Geroch and Kinnersley for gravitational waves (2-dimensional) will be suggested. In particular, they will be recognized as Kac-Moody symmetries and this will help to simplify the equations and hopefully the solution-generating procedures.
The author obtains a number of known exact electrovacuum solutions of the Einstein-Maxwell equations, and also various generalizations of these solutions. Successive application of the transformations described permits one to obtain arbitrary generalizations of the chosen starting solution.
The author describes a generalization of the inverse scattering problem methods to the case of the presence in space not only of gravitational and electromagnetic fields but also of a classical `neutrino' field, i.e., a massless two-component spinor field (Weyl field), which depends, like the other fields, only on two coordinates.
The equations of motion for (D = 4) gravitational plane waves { gmn(t,x1) } m,n = 0,1,2,3 and more generally the sets of classical solutions of supergravities N' = 1,2,...,7 that depend only on 2 of the 4 coordinates are invariant under transformations of Kac-Moody (affine type 1) algebras. These invariances are merely verified by explicit computation, however they form a regular pattern and the table of "internal" symmetries of these supergravities in various dimensions suggests a deeper and wider role for Kac-Moody algebras. We shall provide as many definitions as possible and review the "dimensional reduction" from the dual string model of Ramond, Neveu and Schwarz (d = 10) to plane waves with effective dimension d = 2. In this process, the algebra of symmetries that preserve the effective coordinates (internal symmetries) grows regularly by increasing its real rank. The same property can be observed in some real forms of the magic square of Freudenthal and Tits. The connection with completely integrable systems and in particular non-local charges will be briefly mentioned. Finally we shall describe the converse of dimensional reduction called "group disintegration" (increasing space-time dimension) and make a few comments about relevant superalgebras.
The author offers a further development of the method of the inverse problem for the description of solutions of the general (and not only purely soliton) type, expressed in a complete reformulation of the field equations in terms of a linear system with a complex parameter, with a subsequent transition to the equivalent linear (nonmatrix) singular integral equations, for which the kernel is determined from the initial or boundary conditions, given at an arbitrary chosen smooth boundary. This approach, following from the most elementary considerations, seems the most simple and natural approach and can serve as a basis for the construction of effective methods of studying the general properties of solutions, the statement and solution of physically interesting initial-value and boundary-value problems, the construction of various approximations, or the search for new families of exact solutions.
This paper is made up of two parts. The extended Introduction contains a historical survey showing the role of exact solutions in the development of the theory of gravitation. In the main part, based on a series of articles of the author, new methods are expounded for generating exact solutions of the Einstein-Maxwell equations for interacting gravitational and electromagnetic fields away from their sources. Here it is assumed that all the fields and their potentials depend only on some two space-time coordinates, and hence space-time admits a two-dimensional Abelian group of isometries. A description is given of the generation of families of L-soliton solutions, as well as nonsoliton solutions of the simplest type, starting from an arbitrarily chosen bare background solution. Various concrete examples are given. A new approach to the description of a locally general solution is developed; it leads to an equivalent scalar (i.e., not metric) linear singular integral equation whose kernel contains a complete collection of arbitrary functions parametrizing the locally general solution. These functions are uniquely determined by the initial or boundary conditions in solving various boundary value problems (in particular, the Cauchy problem). Particular examples of solutions are considered.
A new form of the general solution of the initial value problem for colliding gravitational plane waves with collinear polarization is obtained. The solution of the linear hyperbolic field equation for y(u,v) is expressed as a linear superposition int ds g(s) w(u,v,s) of a one-parameter family of basic solutions of the form w(u,v,s) = w1(u,s) w2(v,s), where u and v are arbitrary null coordinates and s is the parameter. The coefficients g(s) in this superposition are expressed in terms of the initial data by using a generalization of an integral transform obtained by Abel in his solution of a tautochrone problem of classical particle mechanics.
As a preliminary step in the development of a Hilbert problem (HP) approach to the initial value problem (IVP) for colliding gravitational plane waves with noncollinear polarizations, the IVP for colliding gravitational plane waves with collinear polarizations is reformulated in two different ways as an HP in a complex plane. The solutions of both forms of the HP are found and each of these agrees with the solution obtained by another method in the previous paper of this series [I. Hauser and F. J. Ernst, J. Math. Phys. 30, 872 (1989)]. The conditions imposed on the initial data of the IVP by the vacuum field equations are discussed in detail. Anticipating the next paper of this series, the generalization of one form of the HP to noncollinear polarizations is briefly described.
The development of a homogeneous Hilbert problem (HHP) approach to the initial value problem (IVP) for colliding gravitational plane waves with noncollinear polarization that began in two earlier papers [I. Hauser and F. J. Ernst, J. Math. Phys. 30, 872 (1989) and 30, 2322 (1989)] is continued. After formulating the HHP, the description of how one can apply it to generate a new family of solutions of the colliding wave problem that generalizaes a three-parameter family constructed by Ernst, García, and Hauser [J. Math. Phys. 29, 681 (1988)] using a double-Harrison transformation is given. Then the proof that the soluiton of the new HHP indeed solves the IVP that is posed is presented. A matrix Fredholm equation of the second kind that is equivalent to the HHP is also deduced. This will be used in a sequel to complete the proof of existence of solutions of the HHP and the proof that certain assumed differentiability hypotheses are in fact valid.
For electrovacuum with two space-like commuting Killing vectors, whose orbits have, respectively, topologies of a straight line and a circle, a linear integral equation is derived for describing fields generated by filamentary sources on the symmetry axis, which have been "burning" for a finite time. The relationship between the asymptotic expansion of field characteristics after the sources have been switched off and the characteristics of time evolution of the source burning intensity is found.
In a preceding paper [I. Hauser and F. J. Ernst, J. Math. Phys. 31, 871 (1990)], the initial value problem for colliding gravitational plane waves was reformulated as a homogeneous Hilbert problem (HHP) in a complex plane. It is now proven that a unique solution of this HHP always exists and that any assumed differentiability or holomorphy properties of the plane-wave metrics imply corresponding properties for the scattered wave metric. This completes the demonstration, initiated in the preceding paper, that the solution of the HHP furnishes the solution of the initial value problem. A Fredholm equation that is equivalent to the HHP and that was introduced in the preceding paper is used to effect the proofs.
Having defined a set S3E that includes all C4 solutions E of the hyperbolic Ernst equation in a certain specified domain, we formulate and prove, using a new HHP and equivalent integral equations of the Alekseev and Fredholm types, a generalized Geroch conjecture concerning the transformation of one member of S3E into another member. (The original Geroch conjecture, as interpreted by Kinnersley and Chitre, concerned the transformation of one analytic solution of the elliptic Ernst equation into another analytic solution.) A unique feature of these notes is that we provide a mathematical proof for every nontrivial statement that we make.
It should be noted that some of the propositions and theorems that are included in these notes were developed before we knew exactly what would be absolutely necessary in order to accomplish our principal objectives most expeditiously.
We enunciate and prove here a generalization of Geroch's famous conjecture concerning analytic solutions of the elliptic Ernst equation. Our generalization is stated for solutions of the hyperbolic Ernst equation that are not necessarily analytic, although it can be formulated also for solutions of the elliptic Ernst equation that are nowhere axis-accessible.
A monodromy transform approach, presented in this communication, provides the general base for solution of space-time symmetry reductions of Einstein's equations in all known integrable cases, which include vacuum, electrovacuum, massless Weyl spinor field and stiff fluid, as well as some string theory induced gravity models. There were found a special finite set of functional parameters which are defined as the set of monodromy data for the fundamental solution of associated spectral problem. These monodromy data consist of the function of the spectral parameter only. Similarly to the scattering data in the inverse scattering transform, the monodromy data can be used for characterization of any local solution of the field equations. The ``direct'' and ``inverse'' problems of such monodromy data transform admit unambiguous solutions. For the linear singular integral equation with a scalar (i.e., non-matrix) kernel, which solves the inverse problem of this monodromy transform, an equivalent regularization --- a Fredholm linear integral equation of the second kind is constructed in several convenient forms. The existence and uniqueness of the local solution for arbitrary choice of monodromy data can be proved using a simple interative method. This solution is effectively constructed in terms of homogeneously convergent functional series.
The approach, referred to as ``monodromy transform'', provides some general base for solution of all known integrable space-time symmetry reductions of Einstein equations for the case of pure vacuum gravitational fields, in the presence of gravitationally interacting massless fields, as well as for some string theory induced gravity models. In this communication we present the key points of this approach, applied to Einstein equations for vacuum and to Einstein-Maxwell equations for electrovacuum fields in the cases, reducible to the known Ernst equations. Definition of the monodromy data, formulation and solution of the direct and inverse problems of the monodromy transform, a proof of existence and uniqueness of their solutions, the structure of the basic linear singular integral equations and their regularizations, which lead to the equations of (quasi-)Fredholm type are also discussed. A construction of general local solution of these equations is given in terms of homogeneously convergent functional series.
In this paper the well known Belinskii and Zakharov soliton generating transformations of the solution space of vacuum Einstein equations with two-dimensional Abelian groups of isometries are considered in the context of the so called ``monodromy transform approach'', which provides some general base for the study of various integrable space-time symmetry reductions of Einstein equations. Similarly to the scattering data used in the known spectral transform, in this approach the monodromy data for solution of associated linear system characterize completely any solution of the reduced Einstein equations, and many physical and geometrical properties of the solutions can be expressed directly in terms of the analytical structure on the spectral plane of the corresponding monodromy data functions. The Belinskii and Zakharov vacuum soliton generating transformations can be expressed in explicit form (without specification of the corresponding monodromy data functions with coefficients, polynomial in spectral parameter. This allows to determine many physical parameters of the generating soliton solutions without (or before) calculation of all components of the solutions. The similar characterization for electrovacuum soliton generating transformations is also presented.
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